How to bake a particle animation over 250 frames
I think I understand what's happening in the video and why this is causing confusion.
For some simulations (such as Rigid Body) you set the range of frames on the cache and Blender will bake all frames within that range.
For particles, things are slightly difference since particles can exist for only a part of the range of frames - so the cache is automatic in that it will store those frames that have particles that are within the range of timeline frames. ie, you do not need to explicitly specify a range of frames to cache.
In the example video, the timeline frames are amended to the range 1 to 500 but the particle system is still defined as having particles emitted from frame 1 to frame 200 with a life of 50 for each particle. This means that the cache actually bakes 251 frames - with the last frame (251) to indicate purely that all particles have died. You can see the frame count in the right-hand panel just after the bake has completed.
So, all you need to do to have 500 frames of particles (after you have amended the timeline frame range end at frame 500) is to change the particle system to emit particles that persist after the 250th frame - eg, increase 'End' to 500 or increase 'Lifetime' to, say, 500. Running or baking the simulation will result in particles for the full frame range.